Analysis: survey
Gamified Feedback in Adaptive Retrieval Practice
Maarten van der Velde & Gesa van den Broek
Last updated: 2024-11-20
Overview
This notebook analyses the survey data collected in the study. After each practice block, participants were asked about task-motivational outcomes (feelings of competence, enjoyment, task value), goal setting behaviour (experienced goal setting, trying to perform well), potential negative effects (stress, distraction), perceived relevance of the form of gamified feedback they encountered in the practice block, and were asked to give a judgement of learning. After the last practice block, participants also indicated their preferred feedback type from the types they had encountered during practice.
This notebook analyses the effects of gamification on these outcomes.
We use mixed-effects regression models with the same set of
theoretically-motivated predictors across all analyses. We report both a
frequentist (using lme4 and lmerTest) and a
Bayesian (using brms) analysis for each outcome.
Setup
library(here)here() starts at /Users/maarten/Documents/projects/gamified-feedback-studylibrary(dplyr)
Attaching package: 'dplyr'The following objects are masked from 'package:stats':
filter, lagThe following objects are masked from 'package:base':
intersect, setdiff, setequal, unionlibrary(ggplot2)
library(scales)
library(patchwork)Warning: package 'patchwork' was built under R version 4.3.3library(stringr)
library(tidyr)
library(lme4)Loading required package: Matrix
Attaching package: 'Matrix'The following objects are masked from 'package:tidyr':
expand, pack, unpacklibrary(lmerTest)
Attaching package: 'lmerTest'The following object is masked from 'package:lme4':
lmerThe following object is masked from 'package:stats':
steplibrary(brms)Loading required package: RcppLoading 'brms' package (version 2.21.0). Useful instructions
can be found by typing help('brms'). A more detailed introduction
to the package is available through vignette('brms_overview').
Attaching package: 'brms'The following object is masked from 'package:lme4':
ngrpsThe following object is masked from 'package:stats':
arlibrary(BayesFactor)Loading required package: coda************
Welcome to BayesFactor 0.9.12-4.7. If you have questions, please contact Richard Morey (richarddmorey@gmail.com).
Type BFManual() to open the manual.
************# Set up parallel processing for Bayesian models
library(future)
plan(multisession, workers = 4)Helper functions for plots and tables:
source(here("scripts", "00_visualisation_functions.R"))Loading required package: data.table
Attaching package: 'data.table'The following objects are masked from 'package:dplyr':
between, first, lastLoading required package: flextableLoad processed data:
d_survey <- readRDS(here("data", "processed", "d_survey.rds"))add_experiment_cols <- function (data) {
data |>
mutate(exp_order = case_when(
gamified_first == 0 & exp_group == "score" ~ "Control—Score",
gamified_first == 0 & exp_group == "both" ~ "Control—Both",
gamified_first == 1 & exp_group == "score" ~ "Score—Control",
gamified_first == 1 & exp_group == "both" ~ "Both—Control"
)) |>
mutate(type = ifelse(gamified, "Gamified", "Control"))
}Helper function for interpreting Bayes factors:
bf_to_strength <- function (bf) {
direction <- "for"
if (bf < 1) {
bf <- 1/bf
direction <- "against"
}
strength <- case_when(
bf == 1 ~ "No",
bf < 3 ~ "Anecdotal",
bf < 10 ~ "Moderate",
bf < 30 ~ "Strong",
bf < 100 ~ "Very strong",
TRUE ~ "Extreme"
)
paste0(strength, " evidence ", direction)
}Does gamification change task-motivational outcomes?
Relevant variables: feelings of competence, enjoyment, task value.
Prepare data
d_survey_agg <- d_survey |>
group_by(block, condition, gamified, gamified_first, exp_group, category, question) |>
summarise(response_mean = mean(response, na.rm = T),
response_se = sd(response, na.rm = T)/sqrt(n())) |>
ungroup() |>
add_experiment_cols() |>
mutate(perception_label_sorted = factor(question, levels = c("goalsetting","performwell","goalstress","distraction","relevance")))`summarise()` has grouped output by 'block', 'condition', 'gamified',
'gamified_first', 'exp_group', 'category'. You can override using the `.groups`
argument.Mean-centering categorical predictors for modelling:
d_survey_m <- d_survey |>
mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
exp_group_c = exp_group_c - mean(exp_group_c),
gamified_first_c = gamified_first - mean(gamified_first))Combined plot
dodge_width <- .25
p_motivation <- d_survey_agg |>
filter(category == "motivation") |>
ggplot(aes(x = block, y = response_mean, group = interaction(exp_order, question))) +
facet_grid(~ question, labeller = labeller(question = str_to_title)) +
geom_line(aes(lty = gamified_first), position = position_dodge(width = dodge_width), lwd = .4) +
geom_errorbar(aes(ymin = response_mean - response_se, ymax = response_mean + response_se, colour = condition),
width = 0,
alpha = .5,
position = position_dodge(width = dodge_width)) +
geom_point(aes(colour = condition, pch = condition),
size = 2,
position = position_dodge(width = dodge_width)) +
scale_y_continuous(breaks = 1:7) +
scale_colour_manual(values = col_condition) +
scale_linetype_manual(values = c(1, 2), labels = c("Control first", "Gamified first")) +
labs(x = "Block",
y = "Response",
colour = "Condition",
pch = "Condition",
lty = "Order") +
theme_paper
p_motivation
ggsave(p_motivation, filename = here("output", "survey_motivation.png"), width = 7.5, height = 3)Competence
Fit frequentist model
m_competence <- lmer(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "competence"))
summary(m_competence)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "competence")
REML criterion at convergence: 1119.5
Scaled residuals:
Min 1Q Median 3Q Max
-2.32430 -0.51027 0.00453 0.51192 2.02550
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 1.2298 1.1090
Residual 0.8614 0.9281
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df
(Intercept) 3.95034 0.11224 240.74459
gamifiedTRUE 0.45649 0.10188 162.00000
gamifiedFALSE:exp_group_c -0.24880 0.22474 240.74459
gamifiedTRUE:exp_group_c 0.11862 0.22474 240.74459
gamifiedFALSE:gamified_first_c -0.13313 0.22528 240.74459
gamifiedTRUE:gamified_first_c -0.19704 0.22528 240.74459
gamifiedFALSE:exp_group_c:gamified_first_c -0.03749 0.45111 240.74459
gamifiedTRUE:exp_group_c:gamified_first_c -0.19749 0.45111 240.74459
t value Pr(>|t|)
(Intercept) 35.195 < 2e-16 ***
gamifiedTRUE 4.481 1.4e-05 ***
gamifiedFALSE:exp_group_c -1.107 0.269
gamifiedTRUE:exp_group_c 0.528 0.598
gamifiedFALSE:gamified_first_c -0.591 0.555
gamifiedTRUE:gamified_first_c -0.875 0.383
gamifiedFALSE:exp_group_c:gamified_first_c -0.083 0.934
gamifiedTRUE:exp_group_c:gamified_first_c -0.438 0.662
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.454
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.588
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.002
gmfdTRUE:g__ 0.000 0.000 -0.002 -0.004 0.588
gFALSE:__:_ -0.004 0.002 0.000 0.000 0.001
gTRUE:__:__ -0.002 -0.002 0.000 0.000 0.000
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.000
gTRUE:__:__ 0.001 0.588 print_model_table(m_competence)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 3.950 | 0.112 | 240.74 | 35.20 | <.001 |
Gamification | 0.456 | 0.102 | 162.00 | 4.48 | <.001 |
Type of gamification (Points-and-progress - Points) | 0.119 | 0.225 | 240.74 | 0.53 | .598 |
Type of gamification on Control (Points-and-progress - Points) | -0.249 | 0.225 | 240.74 | -1.11 | .269 |
Block on gamification (1 - 2) | -0.197 | 0.225 | 240.74 | -0.87 | .383 |
Block on Control (2 - 1) | -0.133 | 0.225 | 240.74 | -0.59 | .555 |
Type of gamification; block difference (Block 1 - Block 2) | -0.197 | 0.451 | 240.74 | -0.44 | .662 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -0.037 | 0.451 | 240.74 | -0.08 | .934 |
Fitted values
d_model_fit <- crossing(
gamified = c(TRUE, FALSE),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_competence,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_competence_m <- plot_model_fit(m_competence, filter(d_survey_m, question == "competence"), y_lab = "Competence") +
scale_y_continuous(limits = c(3, 6), labels = scales::comma_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 3.902778 Control—Both Control
2 -0.4759036 4.131250 Control—Score Control
3 -0.4759036 4.303191 Score—Control Gamified
4 0.5240964 4.331395 Both—Control Gamified
5 0.5240964 3.750000 Both—Control Control
6 -0.4759036 4.015957 Score—Control Control
7 -0.4759036 4.406250 Control—Score Gamified
8 0.5240964 4.631944 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_competence_m
Fit Bayesian model
m_competence_bayes <- brm(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "competence"),
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
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Chain 4: summary(m_competence_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "competence") (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 1.11 0.09 0.95 1.29 1.00 12390 21795
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 3.95 0.11 3.73 4.17
gamifiedTRUE 0.45 0.10 0.25 0.65
gamifiedFALSE:exp_group_c -0.24 0.22 -0.67 0.19
gamifiedTRUE:exp_group_c 0.12 0.22 -0.31 0.54
gamifiedFALSE:gamified_first_c -0.12 0.22 -0.55 0.30
gamifiedTRUE:gamified_first_c -0.19 0.22 -0.62 0.25
gamifiedFALSE:exp_group_c:gamified_first_c -0.01 0.40 -0.81 0.77
gamifiedTRUE:exp_group_c:gamified_first_c -0.16 0.40 -0.95 0.63
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 24434 28437
gamifiedTRUE 1.00 82895 27999
gamifiedFALSE:exp_group_c 1.00 22129 28957
gamifiedTRUE:exp_group_c 1.00 21794 28643
gamifiedFALSE:gamified_first_c 1.00 21855 28028
gamifiedTRUE:gamified_first_c 1.00 22243 28415
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 25998 31069
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 26191 30603
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.94 0.05 0.84 1.05 1.00 17125 26124
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_competence_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_competence <- hypothesis(m_competence_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_competence$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio
at zero, which is a ratio of the posterior density at zero relative to
the prior density at zero (indicated by dashed vertical line). Values
above 1 indicate a stronger belief that the effect is indeed zero after
having observed the data. See Wagenmakers et al. (2010) for details on
this test, and https://mvuorre.github.io/posts/2017-03-21-bayes-factors-with-brms/
for a reference on doing this with brms.
sd_ratio_competence <- plot(h_test_competence, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_competence[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
d_model_fit <- crossing(
gamified = c(TRUE, FALSE),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_competence_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitEnjoyment
Fit frequentist model
m_enjoyment <- lmer(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "enjoyment"))
summary(m_enjoyment)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "enjoyment")
REML criterion at convergence: 923.4
Scaled residuals:
Min 1Q Median 3Q Max
-2.28644 -0.46853 0.04077 0.45253 2.13674
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 0.8818 0.9391
Residual 0.3950 0.6285
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df
(Intercept) 4.48639 0.08770 219.36009
gamifiedTRUE 0.54544 0.06898 162.00000
gamifiedFALSE:exp_group_c -0.30539 0.17561 219.36009
gamifiedTRUE:exp_group_c 0.15226 0.17561 219.36009
gamifiedFALSE:gamified_first_c -0.63243 0.17603 219.36009
gamifiedTRUE:gamified_first_c 0.14920 0.17603 219.36009
gamifiedFALSE:exp_group_c:gamified_first_c -0.83175 0.35248 219.36009
gamifiedTRUE:exp_group_c:gamified_first_c -0.49793 0.35248 219.36009
t value Pr(>|t|)
(Intercept) 51.155 < 2e-16 ***
gamifiedTRUE 7.907 3.86e-13 ***
gamifiedFALSE:exp_group_c -1.739 0.083428 .
gamifiedTRUE:exp_group_c 0.867 0.386862
gamifiedFALSE:gamified_first_c -3.593 0.000404 ***
gamifiedTRUE:gamified_first_c 0.848 0.397584
gamifiedFALSE:exp_group_c:gamified_first_c -2.360 0.019168 *
gamifiedTRUE:exp_group_c:gamified_first_c -1.413 0.159185
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.393
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.691
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.003
gmfdTRUE:g__ 0.000 0.000 -0.003 -0.004 0.691
gFALSE:__:_ -0.004 0.002 0.000 0.000 0.001
gTRUE:__:__ -0.003 -0.002 0.000 0.000 0.000
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.000
gTRUE:__:__ 0.001 0.691 print_model_table(m_enjoyment)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 4.486 | 0.088 | 219.36 | 51.15 | <.001 |
Gamification | 0.545 | 0.069 | 162.00 | 7.91 | <.001 |
Type of gamification (Points-and-progress - Points) | 0.152 | 0.176 | 219.36 | 0.87 | .387 |
Type of gamification on Control (Points-and-progress - Points) | -0.305 | 0.176 | 219.36 | -1.74 | .083 |
Block on gamification (1 - 2) | 0.149 | 0.176 | 219.36 | 0.85 | .398 |
Block on Control (2 - 1) | -0.632 | 0.176 | 219.36 | -3.59 | <.001 |
Type of gamification; block difference (Block 1 - Block 2) | -0.498 | 0.352 | 219.36 | -1.41 | .159 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -0.832 | 0.352 | 219.36 | -2.36 | .019 |
Fitted values
d_model_fit <- crossing(
gamified = c(TRUE, FALSE),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_enjoyment,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitd_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_enjoyment,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitd_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_enjoyment,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_enjoyment_m <- plot_model_fit(m_enjoyment, filter(d_survey_m, question == "enjoyment"), y_lab = "Enjoyment") +
scale_y_continuous(limits = c(3, 6), labels = scales::comma_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 4.905556 Control—Both Control
2 -0.4759036 4.760000 Control—Score Control
3 -0.4759036 5.136170 Score—Control Gamified
4 0.5240964 5.060465 Both—Control Gamified
5 0.5240964 3.837209 Both—Control Control
6 -0.4759036 4.523404 Score—Control Control
7 -0.4759036 4.750000 Control—Score Gamified
8 0.5240964 5.172222 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_enjoyment_m
Fit Bayesian model
m_enjoyment_bayes <- brm(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "enjoyment"),
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: summary(m_enjoyment_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "enjoyment") (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.94 0.07 0.82 1.08 1.00 11228 22789
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 4.49 0.09 4.31 4.66
gamifiedTRUE 0.54 0.07 0.41 0.68
gamifiedFALSE:exp_group_c -0.30 0.17 -0.64 0.04
gamifiedTRUE:exp_group_c 0.16 0.17 -0.18 0.49
gamifiedFALSE:gamified_first_c -0.61 0.17 -0.95 -0.28
gamifiedTRUE:gamified_first_c 0.16 0.17 -0.18 0.50
gamifiedFALSE:exp_group_c:gamified_first_c -0.71 0.33 -1.35 -0.07
gamifiedTRUE:exp_group_c:gamified_first_c -0.39 0.33 -1.03 0.25
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 14172 23030
gamifiedTRUE 1.00 81404 27741
gamifiedFALSE:exp_group_c 1.00 12929 21242
gamifiedTRUE:exp_group_c 1.00 13206 21816
gamifiedFALSE:gamified_first_c 1.00 12698 21773
gamifiedTRUE:gamified_first_c 1.00 13144 21628
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 15905 24017
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 15589 24687
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.63 0.04 0.57 0.71 1.00 16303 24044
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_enjoyment_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_enjoyment <- hypothesis(m_enjoyment_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_enjoyment$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_enjoyment <- plot(h_test_enjoyment, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_enjoyment[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
d_model_fit <- crossing(
gamified = c(TRUE, FALSE),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_enjoyment_bayes,,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitd_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_enjoyment_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitd_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_enjoyment_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitPerceived Task Value
Fit frequentist model
m_value <- lmer(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "value"))
summary(m_value)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "value")
REML criterion at convergence: 911.9
Scaled residuals:
Min 1Q Median 3Q Max
-2.96908 -0.46010 0.04555 0.52620 2.10776
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 0.7933 0.8907
Residual 0.3998 0.6323
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df
(Intercept) 5.43795 0.08478 224.66834
gamifiedTRUE 0.32761 0.06940 162.00000
gamifiedFALSE:exp_group_c -0.43243 0.16976 224.66834
gamifiedTRUE:exp_group_c -0.13120 0.16976 224.66834
gamifiedFALSE:gamified_first_c -0.52788 0.17017 224.66834
gamifiedTRUE:gamified_first_c -0.02039 0.17017 224.66834
gamifiedFALSE:exp_group_c:gamified_first_c -1.18186 0.34074 224.66834
gamifiedTRUE:exp_group_c:gamified_first_c -0.49049 0.34074 224.66834
t value Pr(>|t|)
(Intercept) 64.142 < 2e-16 ***
gamifiedTRUE 4.720 5.06e-06 ***
gamifiedFALSE:exp_group_c -2.547 0.011522 *
gamifiedTRUE:exp_group_c -0.773 0.440397
gamifiedFALSE:gamified_first_c -3.102 0.002168 **
gamifiedTRUE:gamified_first_c -0.120 0.904718
gamifiedFALSE:exp_group_c:gamified_first_c -3.469 0.000627 ***
gamifiedTRUE:exp_group_c:gamified_first_c -1.439 0.151407
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.409
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.665
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.003
gmfdTRUE:g__ 0.000 0.000 -0.003 -0.004 0.665
gFALSE:__:_ -0.004 0.002 0.000 0.000 0.001
gTRUE:__:__ -0.003 -0.002 0.000 0.000 0.000
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.000
gTRUE:__:__ 0.001 0.665 print_model_table(m_value)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 5.438 | 0.085 | 224.67 | 64.14 | <.001 |
Gamification | 0.328 | 0.069 | 162.00 | 4.72 | <.001 |
Type of gamification (Points-and-progress - Points) | -0.131 | 0.170 | 224.67 | -0.77 | .440 |
Type of gamification on Control (Points-and-progress - Points) | -0.432 | 0.170 | 224.67 | -2.55 | .012 |
Block on gamification (1 - 2) | -0.020 | 0.170 | 224.67 | -0.12 | .905 |
Block on Control (2 - 1) | -0.528 | 0.170 | 224.67 | -3.10 | .002 |
Type of gamification; block difference (Block 1 - Block 2) | -0.490 | 0.341 | 224.67 | -1.44 | .151 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -1.182 | 0.341 | 224.67 | -3.47 | .001 |
Fitted values
overall difference in reported task value between gamified and non-gamified conditions…
d_model_fit <- crossing(
gamified = c(TRUE, FALSE),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_value,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitwithin the Control condition, reported task value is higher in the Points experimental group than in the Progress bar group…
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_value,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitwithin the Control condition, reported task value is higher in the Block 1 than in Block 2…
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_value,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitd_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_value,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_value_m <- plot_model_fit(m_value, filter(d_survey_m, question == "value"), y_lab = "Value") +
scale_y_continuous(limits = c(3, 6), labels = scales::comma_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 5.833333 Control—Both Control
2 -0.4759036 5.625000 Control—Score Control
3 -0.4759036 5.925532 Score—Control Gamified
4 0.5240964 5.569767 Both—Control Gamified
5 0.5240964 4.686047 Both—Control Control
6 -0.4759036 5.659574 Score—Control Control
7 -0.4759036 5.712500 Control—Score Gamified
8 0.5240964 5.847222 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_value_m
Fit Bayesian model
m_value_bayes <- brm(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "value"),
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
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Chain 4: summary(m_value_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "value") (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.89 0.06 0.77 1.02 1.00 12641 23192
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 5.44 0.08 5.27 5.60
gamifiedTRUE 0.33 0.07 0.19 0.46
gamifiedFALSE:exp_group_c -0.42 0.17 -0.75 -0.09
gamifiedTRUE:exp_group_c -0.12 0.17 -0.45 0.21
gamifiedFALSE:gamified_first_c -0.51 0.17 -0.84 -0.18
gamifiedTRUE:gamified_first_c -0.01 0.17 -0.34 0.32
gamifiedFALSE:exp_group_c:gamified_first_c -1.04 0.32 -1.66 -0.42
gamifiedTRUE:exp_group_c:gamified_first_c -0.37 0.32 -0.99 0.26
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 17168 25766
gamifiedTRUE 1.00 85609 28264
gamifiedFALSE:exp_group_c 1.00 15850 23576
gamifiedTRUE:exp_group_c 1.00 15771 25040
gamifiedFALSE:gamified_first_c 1.00 15565 25200
gamifiedTRUE:gamified_first_c 1.00 15923 23666
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 17489 26062
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 17837 25694
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.64 0.04 0.57 0.71 1.00 18083 26124
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_value_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_value <- hypothesis(m_value_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_value$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_value <- plot(h_test_value, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_value[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
d_model_fit <- crossing(
gamified = c(TRUE, FALSE),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_value_bayes,,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitd_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_value_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitd_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_value_bayes,,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitd_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_value,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitConclusion
- All three aspects of task motivation (experienced competence, enjoyment, task value) were rated higher in the gamified conditions than in the control condition, but no different between the points and progress bar condition.
- There was also an indirect effect of the manipulation in that the control condition was rated lower when presented after gamified practice than before gamified practice: both enjoyment and perceived task value were worse if the control condition was presented after the gamified condition, and this effect was more pronounced when the control condition was done after the progress bar condition than after the points condition.
Does gamification change goal setting behavior?
Relevant variables: goal use and wanting to perform well.
Combined plot
dodge_width <- .25
p_perception <- d_survey_agg |>
filter(category == "perception") |>
mutate(question_sorted = factor(question, levels = c("goalsetting","performwell","goalstress","distraction","relevance")))|>
ggplot(aes(x = block, y = response_mean, group = interaction(exp_order, question))) +
facet_grid(~ perception_label_sorted, labeller = labeller(question = str_to_title)) +
geom_line(aes(lty = exp_order), position = position_dodge(width = dodge_width)) +
geom_errorbar(aes(ymin = response_mean - response_se, ymax = response_mean + response_se, colour = condition),
width = 0,
alpha = .5,
position = position_dodge(width = dodge_width)) +
geom_point(aes(colour = condition, pch = condition),
size = 2,
position = position_dodge(width = dodge_width)) +
scale_y_continuous(breaks = 1:7) +
scale_colour_manual(values = col_condition) +
guides(lty = "none") +
labs(x = "Block",
y = "Response",
colour = "Condition",
pch = "Condition") +
theme_paper
p_perceptionWarning: Removed 4 rows containing missing values or values outside the scale
range (`geom_line()`).`geom_line()`: Each group consists of only one observation.
ℹ Do you need to adjust the group aesthetic?Warning: Removed 4 rows containing missing values or values outside the scale
range (`geom_point()`).
ggsave(p_perception, file = here("output", "survey_perception.png"), width = 8, height = 3)Warning: Removed 4 rows containing missing values or values outside the scale
range (`geom_line()`).`geom_line()`: Each group consists of only one observation.
ℹ Do you need to adjust the group aesthetic?Warning: Removed 4 rows containing missing values or values outside the scale
range (`geom_point()`).Goalsetting
Fit frequentist model
m_goalsetting <- lmer(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "goalsetting"))
summary(m_goalsetting)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "goalsetting")
REML criterion at convergence: 1230.9
Scaled residuals:
Min 1Q Median 3Q Max
-2.50358 -0.46869 0.09304 0.60013 2.16834
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 1.225 1.107
Residual 1.456 1.207
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df t value
(Intercept) 4.3441 0.1271 268.0605 34.180
gamifiedTRUE 0.9207 0.1325 162.0000 6.951
gamifiedFALSE:exp_group_c -0.5549 0.2545 268.0605 -2.180
gamifiedTRUE:exp_group_c 0.6310 0.2545 268.0605 2.480
gamifiedFALSE:gamified_first_c -0.8693 0.2551 268.0605 -3.408
gamifiedTRUE:gamified_first_c -0.4116 0.2551 268.0605 -1.613
gamifiedFALSE:exp_group_c:gamified_first_c -0.6956 0.5108 268.0605 -1.362
gamifiedTRUE:exp_group_c:gamified_first_c 0.2417 0.5108 268.0605 0.473
Pr(>|t|)
(Intercept) < 2e-16 ***
gamifiedTRUE 8.44e-11 ***
gamifiedFALSE:exp_group_c 0.030097 *
gamifiedTRUE:exp_group_c 0.013764 *
gamifiedFALSE:gamified_first_c 0.000756 ***
gamifiedTRUE:gamified_first_c 0.107846
gamifiedFALSE:exp_group_c:gamified_first_c 0.174446
gamifiedTRUE:exp_group_c:gamified_first_c 0.636460
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.521
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.457
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.002
gmfdTRUE:g__ 0.000 0.000 -0.002 -0.004 0.457
gFALSE:__:_ -0.004 0.002 0.000 0.000 0.001
gTRUE:__:__ -0.002 -0.002 0.000 0.000 0.000
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.000
gTRUE:__:__ 0.001 0.457 print_model_table(m_goalsetting)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 4.344 | 0.127 | 268.06 | 34.18 | <.001 |
Gamification | 0.921 | 0.132 | 162.00 | 6.95 | <.001 |
Type of gamification (Points-and-progress - Points) | 0.631 | 0.254 | 268.06 | 2.48 | .014 |
Type of gamification on Control (Points-and-progress - Points) | -0.555 | 0.254 | 268.06 | -2.18 | .030 |
Block on gamification (1 - 2) | -0.412 | 0.255 | 268.06 | -1.61 | .108 |
Block on Control (2 - 1) | -0.869 | 0.255 | 268.06 | -3.41 | .001 |
Type of gamification; block difference (Block 1 - Block 2) | 0.242 | 0.511 | 268.06 | 0.47 | .636 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -0.696 | 0.511 | 268.06 | -1.36 | .174 |
Fitted values
Gamified versus control:
d_model_fit <- crossing(
gamified = sort(unique(d_survey_m$gamified)),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalsetting,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitPoints group versus progress bar group:
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalsetting,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitControl condition: gamified first vs. gamified second:
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_goalsetting,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitControl condition: progress bar group vs. points group:
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalsetting,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_goalsetting_m <- plot_model_fit(m_goalsetting, filter(d_survey_m, question == "goalsetting"), y_lab = "Goal use") +
scale_y_continuous(limits = c(3, 6), labels = scales::comma_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 4.722222 Control—Both Control
2 -0.4759036 4.900000 Control—Score Control
3 -0.4759036 4.723404 Score—Control Gamified
4 0.5240964 5.465116 Both—Control Gamified
5 0.5240964 3.488372 Both—Control Control
6 -0.4759036 4.361702 Score—Control Control
7 -0.4759036 5.250000 Control—Score Gamified
8 0.5240964 5.750000 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_goalsetting_m
Fit Bayesian model
m_goalsetting_bayes <- brm(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "goalsetting"),
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: summary(m_goalsetting_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "goalsetting") (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 1.10 0.11 0.89 1.32 1.00 11572 20770
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 4.35 0.13 4.10 4.60
gamifiedTRUE 0.90 0.13 0.64 1.16
gamifiedFALSE:exp_group_c -0.54 0.25 -1.02 -0.06
gamifiedTRUE:exp_group_c 0.61 0.25 0.13 1.09
gamifiedFALSE:gamified_first_c -0.81 0.25 -1.29 -0.32
gamifiedTRUE:gamified_first_c -0.36 0.25 -0.84 0.12
gamifiedFALSE:exp_group_c:gamified_first_c -0.57 0.45 -1.45 0.31
gamifiedTRUE:exp_group_c:gamified_first_c 0.24 0.45 -0.63 1.12
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 35453 32902
gamifiedTRUE 1.00 87730 28694
gamifiedFALSE:exp_group_c 1.00 32829 31613
gamifiedTRUE:exp_group_c 1.00 33605 33542
gamifiedFALSE:gamified_first_c 1.00 32135 31989
gamifiedTRUE:gamified_first_c 1.00 32205 32622
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 38737 32318
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 37679 33306
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.22 0.07 1.09 1.36 1.00 16173 25542
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_goalsetting_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_goalsetting <- hypothesis(m_goalsetting_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_goalsetting$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_goalsetting <- plot(h_test_goalsetting, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_goalsetting[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
Gamified versus control:
d_model_fit <- crossing(
gamified = sort(unique(d_survey_m$gamified)),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalsetting_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitPoints group versus progress bar group:
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalsetting_bayes,,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitControl condition: gamified first vs. gamified second:
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_goalsetting_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitControl condition in progress bar group vs. points group:
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalsetting_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitWanting to perform well
Fit frequentist model
m_performwell <- lmer(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "performwell"))
summary(m_performwell)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "performwell")
REML criterion at convergence: 930.6
Scaled residuals:
Min 1Q Median 3Q Max
-3.0045 -0.3379 0.1355 0.5652 1.6591
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 0.5291 0.7274
Residual 0.5530 0.7436
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df
(Intercept) 5.74133 0.08074 261.48308
gamifiedTRUE 0.49956 0.08163 162.00000
gamifiedFALSE:exp_group_c -0.37273 0.16167 261.48308
gamifiedTRUE:exp_group_c 0.23992 0.16167 261.48308
gamifiedFALSE:gamified_first_c -0.59772 0.16206 261.48308
gamifiedTRUE:gamified_first_c 0.17653 0.16206 261.48308
gamifiedFALSE:exp_group_c:gamified_first_c -0.35958 0.32450 261.48308
gamifiedTRUE:exp_group_c:gamified_first_c 0.07363 0.32450 261.48308
t value Pr(>|t|)
(Intercept) 71.109 < 2e-16 ***
gamifiedTRUE 6.120 6.79e-09 ***
gamifiedFALSE:exp_group_c -2.306 0.021920 *
gamifiedTRUE:exp_group_c 1.484 0.139010
gamifiedFALSE:gamified_first_c -3.688 0.000275 ***
gamifiedTRUE:gamified_first_c 1.089 0.277028
gamifiedFALSE:exp_group_c:gamified_first_c -1.108 0.268841
gamifiedTRUE:exp_group_c:gamified_first_c 0.227 0.820686
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.505
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.489
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.002
gmfdTRUE:g__ 0.000 0.000 -0.002 -0.004 0.489
gFALSE:__:_ -0.004 0.002 0.000 0.000 0.001
gTRUE:__:__ -0.002 -0.002 0.000 0.000 0.000
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.000
gTRUE:__:__ 0.001 0.489 print_model_table(m_performwell)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 5.741 | 0.081 | 261.48 | 71.11 | <.001 |
Gamification | 0.500 | 0.082 | 162.00 | 6.12 | <.001 |
Type of gamification (Points-and-progress - Points) | 0.240 | 0.162 | 261.48 | 1.48 | .139 |
Type of gamification on Control (Points-and-progress - Points) | -0.373 | 0.162 | 261.48 | -2.31 | .022 |
Block on gamification (1 - 2) | 0.177 | 0.162 | 261.48 | 1.09 | .277 |
Block on Control (2 - 1) | -0.598 | 0.162 | 261.48 | -3.69 | <.001 |
Type of gamification; block difference (Block 1 - Block 2) | 0.074 | 0.325 | 261.48 | 0.23 | .821 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -0.360 | 0.325 | 261.48 | -1.11 | .269 |
Fitted values
Gamified versus control:
d_model_fit <- crossing(
gamified = sort(unique(d_survey_m$gamified)),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_performwell,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitPoints group versus progress bar group:
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_performwell,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitControl condition: gamified first vs. gamified second:
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_performwell,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitControl condition in progress bar group vs. points group:
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_performwell,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_performwell_m <- plot_model_fit(m_performwell, filter(d_survey_m, question == "performwell"), y_lab = "Wanting to perform well") +
scale_y_continuous(limits = c(4, 7), labels = scales::comma_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 5.972222 Control—Both Control
2 -0.4759036 6.150000 Control—Score Control
3 -0.4759036 6.191489 Score—Control Gamified
4 0.5240964 6.465116 Both—Control Gamified
5 0.5240964 5.186047 Both—Control Control
6 -0.4759036 5.723404 Score—Control Control
7 -0.4759036 6.050000 Control—Score Gamified
8 0.5240964 6.250000 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_performwell_m
Fit Bayesian model
m_performwell_bayes <- brm(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "performwell"),
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
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Chain 4: summary(m_performwell_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "performwell") (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.73 0.07 0.60 0.86 1.00 12164 21483
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 5.74 0.08 5.58 5.90
gamifiedTRUE 0.50 0.08 0.33 0.66
gamifiedFALSE:exp_group_c -0.36 0.16 -0.68 -0.05
gamifiedTRUE:exp_group_c 0.24 0.16 -0.08 0.55
gamifiedFALSE:gamified_first_c -0.59 0.16 -0.90 -0.27
gamifiedTRUE:gamified_first_c 0.18 0.16 -0.14 0.49
gamifiedFALSE:exp_group_c:gamified_first_c -0.33 0.31 -0.93 0.27
gamifiedTRUE:exp_group_c:gamified_first_c 0.08 0.31 -0.53 0.69
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 35625 33562
gamifiedTRUE 1.00 91023 27346
gamifiedFALSE:exp_group_c 1.00 30531 31534
gamifiedTRUE:exp_group_c 1.00 29197 31722
gamifiedFALSE:gamified_first_c 1.00 30532 32902
gamifiedTRUE:gamified_first_c 1.00 31251 31092
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 33267 32065
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 33583 32070
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.75 0.04 0.67 0.84 1.00 16849 26771
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_performwell_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_performwell <- hypothesis(m_performwell_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_performwell$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_performwell <- plot(h_test_performwell, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_performwell[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
Gamified versus control:
d_model_fit <- crossing(
gamified = sort(unique(d_survey_m$gamified)),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_performwell_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitPoints group versus progress bar group:
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_performwell_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitControl condition: gamified first vs. gamified second:
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_performwell_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitControl condition in progress bar group vs. points group:
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_performwell_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitConclusion
- The two aspects of goal setting (feeling like working towards a goal, trying to perform well) were rated higher in the gamified conditions than the control condition.
- Learners felt more like they were working towards a goal with progress bar feedback than with points, though there was no difference in this regard between the two gamified conditions.
- There were also indirect effects: Learners felt less like working towards a goal and tried less to perform well in the control condition when they had started with gamified practice and/or when in the progress bar group.
Are there negative effects of gamified feedback on learners’ experience?
Relevant variables: stress and distraction.
Stress
Fit frequentist model
m_goalstress <- lmer(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "goalstress"))
summary(m_goalstress)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "goalstress")
REML criterion at convergence: 1186.3
Scaled residuals:
Min 1Q Median 3Q Max
-1.84686 -0.53139 -0.03832 0.52102 1.93172
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 1.487 1.219
Residual 1.069 1.034
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df
(Intercept) 3.584645 0.124081 242.064754
gamifiedTRUE 0.367929 0.113479 162.000003
gamifiedFALSE:exp_group_c 0.213467 0.248451 242.064754
gamifiedTRUE:exp_group_c 0.624127 0.248451 242.064754
gamifiedFALSE:gamified_first_c 0.008963 0.249049 242.064754
gamifiedTRUE:gamified_first_c -0.164511 0.249049 242.064754
gamifiedFALSE:exp_group_c:gamified_first_c -0.303062 0.498695 242.064754
gamifiedTRUE:exp_group_c:gamified_first_c -0.754758 0.498695 242.064754
t value Pr(>|t|)
(Intercept) 28.890 < 2e-16 ***
gamifiedTRUE 3.242 0.00144 **
gamifiedFALSE:exp_group_c 0.859 0.39108
gamifiedTRUE:exp_group_c 2.512 0.01265 *
gamifiedFALSE:gamified_first_c 0.036 0.97132
gamifiedTRUE:gamified_first_c -0.661 0.50952
gamifiedFALSE:exp_group_c:gamified_first_c -0.608 0.54395
gamifiedTRUE:exp_group_c:gamified_first_c -1.513 0.13147
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.457
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.582
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.002
gmfdTRUE:g__ 0.000 0.000 -0.002 -0.004 0.582
gFALSE:__:_ -0.004 0.002 0.000 0.000 0.001
gTRUE:__:__ -0.002 -0.002 0.000 0.000 0.000
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.000
gTRUE:__:__ 0.001 0.582 print_model_table(m_goalstress)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 3.585 | 0.124 | 242.06 | 28.89 | <.001 |
Gamification | 0.368 | 0.113 | 162.00 | 3.24 | .001 |
Type of gamification (Points-and-progress - Points) | 0.624 | 0.248 | 242.06 | 2.51 | .013 |
Type of gamification on Control (Points-and-progress - Points) | 0.213 | 0.248 | 242.06 | 0.86 | .391 |
Block on gamification (1 - 2) | -0.165 | 0.249 | 242.06 | -0.66 | .510 |
Block on Control (2 - 1) | 0.009 | 0.249 | 242.06 | 0.04 | .971 |
Type of gamification; block difference (Block 1 - Block 2) | -0.755 | 0.499 | 242.06 | -1.51 | .131 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -0.303 | 0.499 | 242.06 | -0.61 | .544 |
Fitted values
Gamified versus control:
d_model_fit <- crossing(
gamified = sort(unique(d_survey_m$gamified)),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalstress,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitPoints group versus progress bar group:
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalstress,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_goalstress_m <- plot_model_fit(m_goalstress, filter(d_survey_m, question == "goalstress"), y_lab = "Goal stress") +
scale_y_continuous(limits = c(3, 6), labels = scales::comma_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 3.777778 Control—Both Control
2 -0.4759036 3.400000 Control—Score Control
3 -0.4759036 3.744681 Score—Control Gamified
4 0.5240964 4.023256 Both—Control Gamified
5 0.5240964 3.627907 Both—Control Control
6 -0.4759036 3.553191 Score—Control Control
7 -0.4759036 3.550000 Control—Score Gamified
8 0.5240964 4.583333 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_goalstress_m
Fit Bayesian model
m_goalstress_bayes <- brm(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "goalstress"),
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
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Chain 4: summary(m_goalstress_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "goalstress") (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 1.22 0.10 1.04 1.42 1.00 12415 23496
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 3.59 0.12 3.34 3.83
gamifiedTRUE 0.36 0.12 0.13 0.59
gamifiedFALSE:exp_group_c 0.18 0.24 -0.29 0.65
gamifiedTRUE:exp_group_c 0.58 0.24 0.12 1.05
gamifiedFALSE:gamified_first_c 0.01 0.24 -0.46 0.49
gamifiedTRUE:gamified_first_c -0.16 0.24 -0.63 0.32
gamifiedFALSE:exp_group_c:gamified_first_c -0.17 0.44 -1.03 0.69
gamifiedTRUE:exp_group_c:gamified_first_c -0.58 0.44 -1.43 0.28
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 25458 30192
gamifiedTRUE 1.00 85632 26923
gamifiedFALSE:exp_group_c 1.00 24243 30415
gamifiedTRUE:exp_group_c 1.00 23571 29146
gamifiedFALSE:gamified_first_c 1.00 23586 29792
gamifiedTRUE:gamified_first_c 1.00 23747 29359
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 26061 29458
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 26316 30318
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.04 0.06 0.93 1.16 1.00 16593 25095
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_goalstress_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_goalstress <- hypothesis(m_goalstress_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_goalstress$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_goalstress <- plot(h_test_goalstress, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_goalstress[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
Gamified versus control:
d_model_fit <- crossing(
gamified = sort(unique(d_survey_m$gamified)),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalstress_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitPoints group versus progress bar group:
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_goalstress_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitDistraction
Fit frequentist model
m_distraction <- lmer(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "distraction"))
summary(m_distraction)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "distraction")
REML criterion at convergence: 1216.2
Scaled residuals:
Min 1Q Median 3Q Max
-2.0444 -0.6717 -0.1767 0.4219 2.8290
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 0.6679 0.8172
Residual 1.7075 1.3067
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df
(Intercept) 2.946421 0.119623 300.263104
gamifiedTRUE 0.005577 0.143431 161.999985
gamifiedFALSE:exp_group_c 0.565358 0.239524 300.263104
gamifiedTRUE:exp_group_c 0.578774 0.239524 300.263104
gamifiedFALSE:gamified_first_c -0.709256 0.240101 300.263104
gamifiedTRUE:gamified_first_c -0.916469 0.240101 300.263104
gamifiedFALSE:exp_group_c:gamified_first_c -0.627475 0.480778 300.263104
gamifiedTRUE:exp_group_c:gamified_first_c -0.187730 0.480778 300.263104
t value Pr(>|t|)
(Intercept) 24.631 < 2e-16 ***
gamifiedTRUE 0.039 0.969030
gamifiedFALSE:exp_group_c 2.360 0.018898 *
gamifiedTRUE:exp_group_c 2.416 0.016273 *
gamifiedFALSE:gamified_first_c -2.954 0.003385 **
gamifiedTRUE:gamified_first_c -3.817 0.000164 ***
gamifiedFALSE:exp_group_c:gamified_first_c -1.305 0.192849
gamifiedTRUE:exp_group_c:gamified_first_c -0.390 0.696464
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.600
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.281
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.001
gmfdTRUE:g__ 0.000 0.000 -0.001 -0.004 0.281
gFALSE:__:_ -0.004 0.002 0.000 0.000 0.001
gTRUE:__:__ -0.001 -0.002 0.000 0.000 0.000
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.000
gTRUE:__:__ 0.001 0.281 print_model_table(m_distraction)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 2.946 | 0.120 | 300.26 | 24.63 | <.001 |
Gamification | 0.006 | 0.143 | 162.00 | 0.04 | .969 |
Type of gamification (Points-and-progress - Points) | 0.579 | 0.240 | 300.26 | 2.42 | .016 |
Type of gamification on Control (Points-and-progress - Points) | 0.565 | 0.240 | 300.26 | 2.36 | .019 |
Block on gamification (1 - 2) | -0.916 | 0.240 | 300.26 | -3.82 | <.001 |
Block on Control (2 - 1) | -0.709 | 0.240 | 300.26 | -2.95 | .003 |
Type of gamification; block difference (Block 1 - Block 2) | -0.188 | 0.481 | 300.26 | -0.39 | .696 |
Type of gamification on Control; block difference (Block 2 - Block 1) | -0.627 | 0.481 | 300.26 | -1.31 | .193 |
Fitted values
Points versus progress bar condition
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_distraction,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitControl condition in Block 1 versus Block 2
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$exp_group_c))
)
d_model_fit$model_fit <- predict(m_distraction,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitGamified conditions in Block 1 versus Block 2
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_distraction,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_distraction_m <- plot_model_fit(m_distraction, filter(d_survey_m, question == "distraction"), y_lab = "Distraction") +
scale_y_continuous(limits = c(1.5, 4.5), labels = scales::comma_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 3.805556 Control—Both Control
2 -0.4759036 2.900000 Control—Score Control
3 -0.4759036 2.297872 Score—Control Gamified
4 0.5240964 2.790698 Both—Control Gamified
5 0.5240964 2.767442 Both—Control Control
6 -0.4759036 2.489362 Score—Control Control
7 -0.4759036 3.125000 Control—Score Gamified
8 0.5240964 3.805556 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_distraction_m
Fit Bayesian model
m_distraction_bayes <- brm(response ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = filter(d_survey_m, question == "distraction"),
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
make[1]: *** [foo.o] Error 1Start sampling
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Chain 4: summary(m_distraction_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: response ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: filter(d_survey_m, question == "distraction") (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.80 0.13 0.53 1.04 1.00 8756 14110
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 2.94 0.12 2.71 3.18
gamifiedTRUE 0.01 0.14 -0.28 0.29
gamifiedFALSE:exp_group_c 0.52 0.23 0.06 0.98
gamifiedTRUE:exp_group_c 0.54 0.23 0.08 0.99
gamifiedFALSE:gamified_first_c -0.66 0.23 -1.12 -0.19
gamifiedTRUE:gamified_first_c -0.86 0.23 -1.31 -0.40
gamifiedFALSE:exp_group_c:gamified_first_c -0.50 0.43 -1.35 0.36
gamifiedTRUE:exp_group_c:gamified_first_c -0.12 0.43 -0.97 0.72
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 41816 31684
gamifiedTRUE 1.00 74041 28057
gamifiedFALSE:exp_group_c 1.00 44322 33174
gamifiedTRUE:exp_group_c 1.00 44676 32020
gamifiedFALSE:gamified_first_c 1.00 44681 29925
gamifiedTRUE:gamified_first_c 1.00 43616 32526
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 47323 30664
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 47640 32401
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.32 0.08 1.18 1.48 1.00 12790 19233
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_distraction_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_distraction <- hypothesis(m_distraction_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_distraction$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_distraction <- plot(h_test_distraction, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_distraction[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
Points versus progress bar condition
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_distraction_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitControl condition in Block 1 versus Block 2
d_model_fit <- crossing(
gamified = FALSE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$exp_group_c))
)
d_model_fit$model_fit <- predict(m_distraction_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitGamified conditions in Block 1 versus Block 2
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = 0,
gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)
d_model_fit$model_fit <- predict(m_distraction_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitConclusion
- Gamified practice was rated as more stressful than the control condition.
- The progress bar condition was rated as more stressful and more distracting than the points condition. (However, ratings were overall relatively low on the 7-point scale.
- There was also an indirect effect in that the gamified feedback was rated as more distracting (and the timer in the control condition as less distracting) when presented in the second block.
Which Condition do Learners prefer?
Relevant variables: perceived relevance of gamification, preference for condition.
Perceived Relevance
Fit frequentist model
Since perceived relevance was only rated in the gamified conditions, we cannot use the same model as elsewhere. Instead we can use a simpler linear regression:
m_relevance <- lm(response ~ exp_group_c +
gamified_first_c +
gamified_first_c:exp_group_c,
data = filter(d_survey_m, question == "relevance", gamified == TRUE))
summary(m_relevance)
Call:
lm(formula = response ~ exp_group_c + gamified_first_c + gamified_first_c:exp_group_c,
data = filter(d_survey_m, question == "relevance", gamified ==
TRUE))
Residuals:
Min 1Q Median 3Q Max
-4.3101 -0.9782 0.3546 1.0233 2.3546
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.0421 0.1148 43.916 < 2e-16 ***
exp_group_c 0.6529 0.2299 2.840 0.00509 **
gamified_first_c -0.1757 0.2304 -0.762 0.44700
exp_group_c:gamified_first_c 0.0258 0.4614 0.056 0.95548
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.479 on 162 degrees of freedom
Multiple R-squared: 0.05059, Adjusted R-squared: 0.03301
F-statistic: 2.877 on 3 and 162 DF, p-value: 0.03782Fitted values
d_model_fit <- crossing(
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0,
)
d_model_fit$model_fit <- predict(m_relevance,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitFit Bayesian model
m_relevance_bayes <- brm(response ~ exp_group_c +
gamified_first_c +
gamified_first_c:exp_group_c,
data = filter(d_survey_m, question == "relevance", gamified == TRUE),
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
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Chain 4:
Chain 4: Elapsed Time: 0.017 seconds (Warm-up)
Chain 4: 0.155 seconds (Sampling)
Chain 4: 0.172 seconds (Total)
Chain 4: summary(m_relevance_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: response ~ exp_group_c + gamified_first_c + gamified_first_c:exp_group_c
Data: filter(d_survey_m, question == "relevance", gamifi (Number of observations: 166)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
Intercept 5.04 0.11 4.82 5.27 1.00 51239
exp_group_c 0.62 0.23 0.17 1.06 1.00 48004
gamified_first_c -0.17 0.23 -0.61 0.28 1.00 46859
exp_group_c:gamified_first_c 0.02 0.42 -0.80 0.84 1.00 51349
Tail_ESS
Intercept 30763
exp_group_c 31074
gamified_first_c 30398
exp_group_c:gamified_first_c 30691
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.49 0.08 1.33 1.66 1.00 46955 30085
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_relevance_bayes, nvariables = 4, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_relevance <- hypothesis(m_relevance_bayes,
c("exp_group_c = 0",
"gamified_first_c = 0",
"exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_relevance$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_relevance <- plot(h_test_relevance, nvariables = 4, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_relevance[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Fitted values
d_model_fit <- crossing(
exp_group_c = sort(unique(d_survey_m$exp_group_c)),
gamified_first_c = 0,
)
d_model_fit$model_fit <- predict(m_relevance_bayes,
newdata = d_model_fit,
re_formula = NA,
type = "response")[,1]
d_model_fitPreference for Condition
Contingency table: what proportion of participants in each condition preferred the gamified practice over the control condition or indicated no preference?
data_preferences <- d_survey |>
filter(preference!="" & condition != "Control") |> # Exclude two participants who did not respond to this item and keep only one row per participant
group_by(exp_group = as.factor(exp_group), subject) |>
summarise(preference = as.factor(first(preference)), .groups = "drop")
# Counts
crosstab_preferences <- xtabs(~exp_group + preference, data = data_preferences)
crosstab_preferences preference
exp_group Experimental NoPreference Control
both 54 7 16
score 51 14 22# Proportions
prop.table(crosstab_preferences,1) preference
exp_group Experimental NoPreference Control
both 0.70129870 0.09090909 0.20779221
score 0.58620690 0.16091954 0.25287356Frequentist Chi-square test
chisq.test(crosstab_preferences)
Pearson's Chi-squared test
data: crosstab_preferences
X-squared = 2.7669, df = 2, p-value = 0.2507Bayesian Chi-square test
bf_preferences <- contingencyTableBF(crosstab_preferences, sampleType = "indepMulti", fixedMargin = "rows")
1/bf_preferencesBayes factor analysis
--------------
[1] Indep. (a=1) : 6.405333 ±0%
Against denominator:
Alternative, non-independence, a = 1
---
Bayes factor type: BFcontingencyTable, independent multinomialConclusions
- Progress bar condition was rated as more relevant than the points condition.
- However, participants in the progress bar group did not show a stronger preference for gamified practice over the control condition than participants in the points group.
Does gamification change learners’ metacognitive judgements?
Relevant variable: judgement of learning.
Judgement of learning
Participants were asked to give the percentage of practiced translations they thought they would still know in two days.
d_jol_agg <- d_survey |>
group_by(subject, block, condition, gamified, gamified_first, exp_group) |>
summarise(judgement_of_learning = judgement_of_learning[1]/100) |>
group_by(block, condition, gamified, gamified_first, exp_group) |>
summarise(jol_mean = mean(judgement_of_learning, na.rm = T),
jol_se = sd(judgement_of_learning, na.rm = T)/sqrt(n())) |>
ungroup() |>
add_experiment_cols()`summarise()` has grouped output by 'subject', 'block', 'condition', 'gamified', 'gamified_first'. You can override using the `.groups` argument.
`summarise()` has grouped output by 'block', 'condition', 'gamified', 'gamified_first'. You can override using the `.groups` argument.dodge_width <- .25
p_jol <- ggplot(d_jol_agg, aes(x = block, y = jol_mean, group = exp_order)) +
geom_line(aes(lty = exp_order), position = position_dodge(width = dodge_width)) +
geom_errorbar(aes(ymin = jol_mean - jol_se, ymax = jol_mean + jol_se, colour = condition),
width = 0,
alpha = .5,
position = position_dodge(width = dodge_width)) +
geom_point(aes(colour = condition, pch = condition),
size = 2,
position = position_dodge(width = dodge_width)) +
scale_colour_manual(values = col_condition) +
scale_y_continuous(labels = scales::percent_format()) +
guides(lty = "none") +
labs(x = "Block",
y = "Judgement of learning",
colour = "Condition",
pch = "Condition") +
theme_paper
p_jol
ggsave(p_jol, filename = here("output", "survey_judgementoflearning.png"), width = 8, height = 3)Fit frequentist model
d_jol_m <- d_survey_m |>
group_by(subject, gamified, block, condition, gamified_first, exp_group, gamified_first_c, exp_group_c) |>
summarise(judgement_of_learning = judgement_of_learning[1]/100) |>
ungroup()`summarise()` has grouped output by 'subject', 'gamified', 'block',
'condition', 'gamified_first', 'exp_group', 'gamified_first_c'. You can
override using the `.groups` argument.m_jol <- lmer(judgement_of_learning ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = d_jol_m)
summary(m_jol)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
judgement_of_learning ~ gamified + gamified:exp_group_c + gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: d_jol_m
REML criterion at convergence: -287.3
Scaled residuals:
Min 1Q Median 3Q Max
-3.6291 -0.4202 -0.0289 0.4106 3.4697
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 0.02623 0.1620
Residual 0.00801 0.0895
Number of obs: 332, groups: subject, 166
Fixed effects:
Estimate Std. Error df
(Intercept) 0.345647 0.014362 204.178850
gamifiedTRUE 0.020967 0.009824 162.000003
gamifiedFALSE:exp_group_c -0.054810 0.028757 204.178850
gamifiedTRUE:exp_group_c -0.035268 0.028757 204.178850
gamifiedFALSE:gamified_first_c -0.048299 0.028826 204.178850
gamifiedTRUE:gamified_first_c -0.019907 0.028826 204.178850
gamifiedFALSE:exp_group_c:gamified_first_c 0.015823 0.057721 204.178850
gamifiedTRUE:exp_group_c:gamified_first_c 0.012826 0.057721 204.178850
t value Pr(>|t|)
(Intercept) 24.067 <2e-16 ***
gamifiedTRUE 2.134 0.0343 *
gamifiedFALSE:exp_group_c -1.906 0.0581 .
gamifiedTRUE:exp_group_c -1.226 0.2214
gamifiedFALSE:gamified_first_c -1.676 0.0954 .
gamifiedTRUE:gamified_first_c -0.691 0.4906
gamifiedFALSE:exp_group_c:gamified_first_c 0.274 0.7843
gamifiedTRUE:exp_group_c:gamified_first_c 0.222 0.8244
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) gmTRUE gmfdFALSE:x__ gmfdTRUE:x__ gmfdFALSE:g__
gamifidTRUE -0.342
gmfdFALSE:x__ 0.000 0.000
gmfdTRUE:x__ 0.000 0.000 0.766
gmfdFALSE:g__ 0.000 0.000 -0.004 -0.003
gmfdTRUE:g__ 0.000 0.000 -0.003 -0.004 0.766
gFALSE:__:_ -0.004 0.001 0.000 0.000 0.001
gTRUE:__:__ -0.003 -0.001 0.000 0.000 0.001
gmfdTRUE:g__ gFALSE:__:
gamifidTRUE
gmfdFALSE:x__
gmfdTRUE:x__
gmfdFALSE:g__
gmfdTRUE:g__
gFALSE:__:_ 0.001
gTRUE:__:__ 0.001 0.766 print_model_table(m_jol)Effect | Estimate | SE | df | t-value | p-value |
|---|---|---|---|---|---|
Intercept (Control) | 0.346 | 0.014 | 204.18 | 24.07 | <.001 |
Gamification | 0.021 | 0.010 | 162.00 | 2.13 | .034 |
Type of gamification (Points-and-progress - Points) | -0.035 | 0.029 | 204.18 | -1.23 | .221 |
Type of gamification on Control (Points-and-progress - Points) | -0.055 | 0.029 | 204.18 | -1.91 | .058 |
Block on gamification (1 - 2) | -0.020 | 0.029 | 204.18 | -0.69 | .491 |
Block on Control (2 - 1) | -0.048 | 0.029 | 204.18 | -1.68 | .095 |
Type of gamification; block difference (Block 1 - Block 2) | 0.013 | 0.058 | 204.18 | 0.22 | .824 |
Type of gamification on Control; block difference (Block 2 - Block 1) | 0.016 | 0.058 | 204.18 | 0.27 | .784 |
Fitted values
Gamified versus control:
d_model_fit <- crossing(
gamified = sort(unique(d_jol_m$gamified)),
exp_group_c = 0,
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_jol,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitPoints group versus progress bar group:
d_model_fit <- crossing(
gamified = TRUE,
exp_group_c = sort(unique(d_jol_m$exp_group_c)),
gamified_first_c = 0
)
d_model_fit$model_fit <- predict(m_jol,
newdata = d_model_fit,
re.form = NA,
type = "response")
d_model_fitVisualise fitted model
p_jol_m <- plot_model_fit(m_jol, d_jol_m, y_lab = "Judgement of learning") |
scale_y_continuous(limits = c(.25, .45), labels = scales::percent_format()) block condition gamified gamified_first exp_group gamified_first_c
1 1 Control FALSE FALSE both -0.5421687
2 1 Control FALSE FALSE score -0.5421687
3 1 Points TRUE TRUE score 0.4578313
4 1 Points-and-progress TRUE TRUE both 0.4578313
5 2 Control FALSE TRUE both 0.4578313
6 2 Control FALSE TRUE score 0.4578313
7 2 Points TRUE FALSE score -0.5421687
8 2 Points-and-progress TRUE FALSE both -0.5421687
exp_group_c pred_val exp_order type
1 0.5240964 0.3386111 Control—Both Control
2 -0.4759036 0.4020000 Control—Score Control
3 -0.4759036 0.3714894 Score—Control Gamified
4 0.5240964 0.3420930 Both—Control Gamified
5 0.5240964 0.2986047 Both—Control Control
6 -0.4759036 0.3461702 Score—Control Control
7 -0.4759036 0.3975000 Control—Score Gamified
8 0.5240964 0.3552778 Control—Both GamifiedScale for y is already present.
Adding another scale for y, which will replace the existing scale.p_jol_m
Fit Bayesian model
m_jol_bayes <- brm(judgement_of_learning ~ gamified +
gamified:exp_group_c +
gamified:gamified_first_c +
gamified:gamified_first_c:exp_group_c +
(1 | subject),
data = d_jol_m,
prior = set_prior("normal(0, 1)", class = "b"),
chains = 4,
iter = 11000,
warmup = 1000,
sample_prior = TRUE,
future = TRUE,
seed = 0)Compiling Stan program...Trying to compile a simple C fileRunning /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 15.0.0 (clang-1500.3.9.4)’
using SDK: ‘MacOSX14.4.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:19:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: 'cmath' file not found
#include <cmath>
^~~~~~~
1 error generated.
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Chain 4: summary(m_jol_bayes) Family: gaussian
Links: mu = identity; sigma = identity
Formula: judgement_of_learning ~ gamified + gamified:exp_group_c + gamified:gamified_first_c + gamified:gamified_first_c:exp_group_c + (1 | subject)
Data: d_jol_m (Number of observations: 332)
Draws: 4 chains, each with iter = 11000; warmup = 1000; thin = 1;
total post-warmup draws = 40000
Multilevel Hyperparameters:
~subject (Number of levels: 166)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.16 0.01 0.14 0.19 1.00 9773 18321
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI
Intercept 0.35 0.01 0.32 0.37
gamifiedTRUE 0.02 0.01 0.00 0.04
gamifiedFALSE:exp_group_c -0.05 0.03 -0.11 0.00
gamifiedTRUE:exp_group_c -0.04 0.03 -0.09 0.02
gamifiedFALSE:gamified_first_c -0.05 0.03 -0.11 0.01
gamifiedTRUE:gamified_first_c -0.02 0.03 -0.08 0.04
gamifiedFALSE:exp_group_c:gamified_first_c 0.02 0.06 -0.10 0.13
gamifiedTRUE:exp_group_c:gamified_first_c 0.01 0.06 -0.10 0.13
Rhat Bulk_ESS Tail_ESS
Intercept 1.00 8770 17201
gamifiedTRUE 1.00 73137 27270
gamifiedFALSE:exp_group_c 1.00 7925 15447
gamifiedTRUE:exp_group_c 1.00 7732 15306
gamifiedFALSE:gamified_first_c 1.00 7859 14730
gamifiedTRUE:gamified_first_c 1.00 7811 13786
gamifiedFALSE:exp_group_c:gamified_first_c 1.00 7394 14880
gamifiedTRUE:exp_group_c:gamified_first_c 1.00 7468 15569
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.09 0.01 0.08 0.10 1.00 18169 26837
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Inspect the posterior sample distributions of the fixed effects:
plot(m_jol_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
Bayes factors
Do a hypothesis test for all fixed-effect coefficients (both main
effects and interactions) in the model being equal to zero. The column
Evid.Ratio shows the Bayes Factor in favour of the null
hypothesis (\(BF_{01}\)).
h_test_jol <- hypothesis(m_jol_bayes,
c("gamifiedTRUE = 0",
"gamifiedFALSE:exp_group_c = 0",
"gamifiedTRUE:exp_group_c = 0",
"gamifiedFALSE:gamified_first_c = 0",
"gamifiedTRUE:gamified_first_c = 0",
"gamifiedFALSE:exp_group_c:gamified_first_c = 0",
"gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
class = "b")
h_test_jol$hypothesis |>
mutate(BF10 = 1 / Evid.Ratio,
evidence_for_null = sapply(Evid.Ratio, bf_to_strength))This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line). Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
sd_ratio_jol <- plot(h_test_jol, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)
sd_ratio_jol[[1]] +
geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
Conclusion
- The frequentist analysis finds that judgements of learning were higher in the gamified conditions than in the control condition, but not significantly different between the points and progress bar condition. The Bayesian analysis finds evidence against all effects.
Save hypothesis testing output for visualisation
fwrite(h_test_competence$hypothesis, here("output", "hypothesis_tests", "h_test_competence.csv"))
fwrite(h_test_enjoyment$hypothesis, here("output", "hypothesis_tests", "h_test_enjoyment.csv"))
fwrite(h_test_value$hypothesis, here("output", "hypothesis_tests", "h_test_value.csv"))
fwrite(h_test_goalsetting$hypothesis, here("output", "hypothesis_tests", "h_test_goalsetting.csv"))
fwrite(h_test_performwell$hypothesis, here("output", "hypothesis_tests", "h_test_performwell.csv"))
fwrite(h_test_goalstress$hypothesis, here("output", "hypothesis_tests", "h_test_goalstress.csv"))
fwrite(h_test_distraction$hypothesis, here("output", "hypothesis_tests", "h_test_distraction.csv"))
fwrite(h_test_relevance$hypothesis, here("output", "hypothesis_tests", "h_test_relevance.csv"))
fwrite(h_test_jol$hypothesis, here("output", "hypothesis_tests", "h_test_jol.csv"))Session info
sessionInfo()R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.5
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: Europe/Amsterdam
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] flextable_0.9.6 data.table_1.14.8 future_1.33.0
[4] BayesFactor_0.9.12-4.7 coda_0.19-4 brms_2.21.0
[7] Rcpp_1.0.11 lmerTest_3.1-3 lme4_1.1-34
[10] Matrix_1.6-1.1 tidyr_1.3.0 stringr_1.5.0
[13] patchwork_1.3.0 scales_1.3.0 ggplot2_3.5.0
[16] dplyr_1.1.3 here_1.0.1
loaded via a namespace (and not attached):
[1] tensorA_0.36.2.1 rstudioapi_0.15.0 jsonlite_1.8.7
[4] magrittr_2.0.3 TH.data_1.1-2 estimability_1.4.1
[7] farver_2.1.1 nloptr_2.0.3 rmarkdown_2.25
[10] ragg_1.2.7 vctrs_0.6.3 minqa_1.2.6
[13] askpass_1.2.0 htmltools_0.5.6 distributional_0.4.0
[16] curl_5.1.0 sass_0.4.7 parallelly_1.36.0
[19] StanHeaders_2.32.6 bslib_0.5.1 plyr_1.8.9
[22] sandwich_3.0-2 emmeans_1.8.9 zoo_1.8-12
[25] cachem_1.0.8 uuid_1.2-0 mime_0.12
[28] lifecycle_1.0.3 pkgconfig_2.0.3 R6_2.5.1
[31] fastmap_1.1.1 shiny_1.8.0 digest_0.6.33
[34] numDeriv_2016.8-1.1 colorspace_2.1-0 ps_1.7.5
[37] rprojroot_2.0.3 textshaping_0.3.7 labeling_0.4.3
[40] fansi_1.0.4 abind_1.4-5 compiler_4.3.1
[43] fontquiver_0.2.1 withr_2.5.1 backports_1.4.1
[46] inline_0.3.19 QuickJSR_1.1.3 pkgbuild_1.4.2
[49] MASS_7.3-60 openssl_2.1.1 gfonts_0.2.0
[52] loo_2.7.0 tools_4.3.1 zip_2.3.1
[55] httpuv_1.6.12 glue_1.6.2 callr_3.7.3
[58] nlme_3.1-163 promises_1.2.1 grid_4.3.1
[61] checkmate_2.3.1 reshape2_1.4.4 generics_0.1.3
[64] gtable_0.3.4 xml2_1.3.5 utf8_1.2.3
[67] pillar_1.9.0 posterior_1.5.0 later_1.3.1
[70] splines_4.3.1 lattice_0.21-9 survival_3.5-7
[73] tidyselect_1.2.0 fontLiberation_0.1.0 pbapply_1.7-2
[76] knitr_1.44 fontBitstreamVera_0.1.1 gridExtra_2.3
[79] V8_4.3.3 crul_1.4.2 stats4_4.3.1
[82] xfun_0.40 bridgesampling_1.1-2 matrixStats_1.3.0
[85] rstan_2.32.6 stringi_1.7.12 yaml_2.3.7
[88] boot_1.3-28.1 evaluate_0.22 codetools_0.2-19
[91] httpcode_0.3.0 officer_0.6.6 gdtools_0.3.7
[94] tibble_3.2.1 cli_3.6.1 RcppParallel_5.1.7
[97] xtable_1.8-4 systemfonts_1.0.4 munsell_0.5.0
[100] processx_3.8.2 jquerylib_0.1.4 globals_0.16.2
[103] parallel_4.3.1 rstantools_2.4.0 MatrixModels_0.5-2
[106] ellipsis_0.3.2 prettyunits_1.2.0 bayesplot_1.11.1
[109] Brobdingnag_1.2-9 listenv_0.9.0 mvtnorm_1.2-3
[112] purrr_1.0.2 crayon_1.5.2 rlang_1.1.1
[115] multcomp_1.4-25 ---
title: "Analysis: survey" 
subtitle: "Gamified Feedback in Adaptive Retrieval Practice"
author: "Maarten van der Velde & Gesa van den Broek"
date: "Last updated: `r Sys.Date()`"
output:
  html_notebook:
    smart: no
    toc: yes
    toc_float: yes
  github_document:
    toc: yes
editor_options: 
  chunk_output_type: inline
---

```{r setup, include = FALSE}
# Remove to disable caching
knitr::opts_chunk$set(cache = TRUE)
```

# Overview

This notebook analyses the survey data collected in the study.
After each practice block, participants were asked about task-motivational outcomes (feelings of competence, enjoyment, task value), goal setting behaviour (experienced goal setting, trying to perform well), potential negative effects (stress, distraction), perceived relevance of the form of gamified feedback they encountered in the practice block, and were asked to give a judgement of learning.
After the last practice block, participants also indicated their preferred feedback type from the types they had encountered during practice.

This notebook analyses the effects of gamification on these outcomes.
We use mixed-effects regression models with the same set of theoretically-motivated predictors across all analyses.
We report both a frequentist (using `lme4` and `lmerTest`) and a Bayesian (using `brms`) analysis for each outcome.

# Setup

```{r}
library(here)
library(dplyr)
library(ggplot2)
library(scales)
library(patchwork)
library(stringr)
library(tidyr)
library(lme4)
library(lmerTest)
library(brms)
library(BayesFactor)

# Set up parallel processing for Bayesian models
library(future)
plan(multisession, workers = 4)
```

Helper functions for plots and tables:
```{r}
source(here("scripts", "00_visualisation_functions.R"))
```

Load processed data:
```{r}
d_survey <- readRDS(here("data", "processed", "d_survey.rds"))
```

```{r}
add_experiment_cols <- function (data) {
  data |>
    mutate(exp_order = case_when(
      gamified_first == 0 & exp_group == "score" ~ "Control—Score",
      gamified_first == 0 & exp_group == "both" ~ "Control—Both",
      gamified_first == 1 & exp_group == "score" ~ "Score—Control",
      gamified_first == 1 & exp_group == "both" ~ "Both—Control"
    )) |>
    mutate(type = ifelse(gamified, "Gamified", "Control"))
}
```

Helper function for interpreting Bayes factors:
```{r}
bf_to_strength <- function (bf) {
  
  direction <- "for"
  
  if (bf < 1) {
    bf <- 1/bf
    direction <- "against"
  }
  
  strength <- case_when(
    bf == 1 ~ "No",
    bf < 3 ~ "Anecdotal",
    bf < 10 ~ "Moderate",
    bf < 30 ~ "Strong",
    bf < 100 ~ "Very strong",
    TRUE ~ "Extreme"
  )
  
  paste0(strength, " evidence ", direction)
}
```


# Does gamification change task-motivational outcomes?

Relevant variables: feelings of competence, enjoyment, task value.

Prepare data

```{r}
d_survey_agg <- d_survey |>
  group_by(block, condition, gamified, gamified_first, exp_group, category, question) |>
  summarise(response_mean = mean(response, na.rm = T),
            response_se = sd(response, na.rm = T)/sqrt(n())) |>
  ungroup() |>
  add_experiment_cols() |>
  mutate(perception_label_sorted = factor(question, levels = c("goalsetting","performwell","goalstress","distraction","relevance")))
```

Mean-centering categorical predictors for modelling:
```{r}
d_survey_m <- d_survey |>
  mutate(exp_group_c = ifelse(exp_group == "score", 0, 1),
         exp_group_c = exp_group_c - mean(exp_group_c),
         gamified_first_c = gamified_first - mean(gamified_first))
```


## Combined plot

```{r}
dodge_width <- .25

p_motivation <- d_survey_agg |>
  filter(category == "motivation") |>
  ggplot(aes(x = block, y = response_mean, group = interaction(exp_order, question))) +
  facet_grid(~ question, labeller = labeller(question = str_to_title)) +
  geom_line(aes(lty = gamified_first), position = position_dodge(width = dodge_width), lwd = .4) +
  geom_errorbar(aes(ymin = response_mean - response_se, ymax = response_mean + response_se, colour = condition),
                width = 0,
                alpha = .5,
                position = position_dodge(width = dodge_width)) +
  geom_point(aes(colour = condition, pch = condition),
             size = 2,
             position = position_dodge(width = dodge_width)) +
  scale_y_continuous(breaks = 1:7) +
  scale_colour_manual(values = col_condition) +
  scale_linetype_manual(values = c(1, 2), labels = c("Control first", "Gamified first")) +
  labs(x = "Block",
       y = "Response",
       colour = "Condition",
       pch = "Condition",
       lty = "Order") +
  theme_paper

p_motivation
ggsave(p_motivation, filename = here("output", "survey_motivation.png"), width = 7.5, height = 3)
```


## Competence

### Fit frequentist model

```{r}
m_competence <- lmer(response ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject),
                     data = filter(d_survey_m, question == "competence"))

summary(m_competence)
print_model_table(m_competence)
```

#### Fitted values

```{r}
d_model_fit <- crossing(
  gamified = c(TRUE, FALSE), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_competence,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```


#### Visualise fitted model

```{r}
p_competence_m <- plot_model_fit(m_competence, filter(d_survey_m, question == "competence"), y_lab = "Competence") +
  scale_y_continuous(limits = c(3, 6), labels = scales::comma_format())

p_competence_m
```
### Fit Bayesian model

```{r}
m_competence_bayes <- brm(response ~ gamified +
                            gamified:exp_group_c +
                            gamified:gamified_first_c +
                            gamified:gamified_first_c:exp_group_c +
                            (1 | subject),
                          data = filter(d_survey_m, question == "competence"),
                          prior = set_prior("normal(0, 1)", class = "b"),
                          chains = 4,
                          iter = 11000,
                          warmup = 1000,
                          sample_prior = TRUE,
                          future = TRUE,
                          seed = 0)

summary(m_competence_bayes)
```
Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_competence_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```
#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_competence <- hypothesis(m_competence_bayes,
                                c("gamifiedTRUE = 0",
                                  "gamifiedFALSE:exp_group_c = 0",
                                  "gamifiedTRUE:exp_group_c = 0",
                                  "gamifiedFALSE:gamified_first_c = 0",
                                  "gamifiedTRUE:gamified_first_c = 0",
                                  "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                                  "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                                class = "b")

h_test_competence$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.
See Wagenmakers et al. (2010) for details on this test, and https://mvuorre.github.io/posts/2017-03-21-bayes-factors-with-brms/ for a reference on doing this with `brms`.


```{r, fig.height = 12, fig.width = 6}
sd_ratio_competence <- plot(h_test_competence, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_competence[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values

```{r}
d_model_fit <- crossing(
  gamified = c(TRUE, FALSE), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_competence_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```



## Enjoyment

### Fit frequentist model 
```{r}
m_enjoyment <- lmer(response ~ gamified +
                      gamified:exp_group_c +
                      gamified:gamified_first_c +
                      gamified:gamified_first_c:exp_group_c +
                      (1 | subject),
                    data = filter(d_survey_m, question == "enjoyment"))

summary(m_enjoyment)
print_model_table(m_enjoyment)
```

#### Fitted values
```{r}
d_model_fit <- crossing(
  gamified = c(TRUE, FALSE), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_enjoyment,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_enjoyment,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_enjoyment,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

#### Visualise fitted model

```{r}
p_enjoyment_m <- plot_model_fit(m_enjoyment, filter(d_survey_m, question == "enjoyment"), y_lab = "Enjoyment") +
  scale_y_continuous(limits = c(3, 6), labels = scales::comma_format())

p_enjoyment_m
```

### Fit Bayesian model

```{r}
m_enjoyment_bayes <- brm(response ~ gamified +
                           gamified:exp_group_c +
                           gamified:gamified_first_c +
                           gamified:gamified_first_c:exp_group_c +
                           (1 | subject),
                         data = filter(d_survey_m, question == "enjoyment"),
                         prior = set_prior("normal(0, 1)", class = "b"),
                         chains = 4,
                         iter = 11000,
                         warmup = 1000,
                         sample_prior = TRUE,
                         future = TRUE,
                         seed = 0)

summary(m_enjoyment_bayes)
```

Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_enjoyment_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```

#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_enjoyment <- hypothesis(m_enjoyment_bayes,
                               c("gamifiedTRUE = 0",
                                 "gamifiedFALSE:exp_group_c = 0",
                                 "gamifiedTRUE:exp_group_c = 0",
                                 "gamifiedFALSE:gamified_first_c = 0",
                                 "gamifiedTRUE:gamified_first_c = 0",
                                 "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                                 "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                               class = "b")

h_test_enjoyment$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.



```{r, fig.height = 12, fig.width = 6}
sd_ratio_enjoyment <- plot(h_test_enjoyment, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_enjoyment[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values
```{r}
d_model_fit <- crossing(
  gamified = c(TRUE, FALSE), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_enjoyment_bayes,,
                                 newdata = d_model_fit,
                                 re_formula = NA, 
                                 type = "response")[,1]

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_enjoyment_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_enjoyment_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```


## Perceived Task Value

### Fit frequentist model

```{r}
m_value <- lmer(response ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject),
                     data = filter(d_survey_m, question == "value"))

summary(m_value)
print_model_table(m_value)
```

#### Fitted values

overall difference in reported task value between gamified and non-gamified conditions...
```{r}
d_model_fit <- crossing(
  gamified = c(TRUE, FALSE), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_value,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

within the Control condition, reported task value is higher in the Points experimental group than in the Progress bar group...
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_value,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

within the Control condition, reported task value is higher in the Block 1 than in Block 2...
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_value,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_value,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

#### Visualise fitted model

```{r}
p_value_m <- plot_model_fit(m_value, filter(d_survey_m, question == "value"), y_lab = "Value") +
  scale_y_continuous(limits = c(3, 6), labels = scales::comma_format())

p_value_m
```
### Fit Bayesian model

```{r}
m_value_bayes <- brm(response ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject),
                     data = filter(d_survey_m, question == "value"),
                     prior = set_prior("normal(0, 1)", class = "b"),
                     chains = 4,
                     iter = 11000,
                     warmup = 1000,
                     sample_prior = TRUE,
                     future = TRUE,
                     seed = 0)

summary(m_value_bayes)
```
Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_value_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```
#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_value <- hypothesis(m_value_bayes,
                           c("gamifiedTRUE = 0",
                             "gamifiedFALSE:exp_group_c = 0",
                             "gamifiedTRUE:exp_group_c = 0",
                             "gamifiedFALSE:gamified_first_c = 0",
                             "gamifiedTRUE:gamified_first_c = 0",
                             "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                             "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                           class = "b")

h_test_value$hypothesis |>
    mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.


```{r, fig.height = 12, fig.width = 6}
sd_ratio_value <- plot(h_test_value, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_value[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values

```{r}
d_model_fit <- crossing(
  gamified = c(TRUE, FALSE), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_value_bayes,,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_value_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_value_bayes,,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_value,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```
## Conclusion

-	All three aspects of task motivation (experienced competence, enjoyment, task value) were rated higher in the gamified conditions than in the control condition, but no different between the points and progress bar condition.
-	There was also an indirect effect of the manipulation in that the control condition was rated lower when presented after gamified practice than before gamified practice: both enjoyment and perceived task value were worse if the control condition was presented after the gamified condition, and this effect was more pronounced when the control condition was done after the progress bar condition than after the points condition.



# Does gamification change goal setting behavior?

Relevant variables: goal use and wanting to perform well. 

## Combined plot

```{r}
dodge_width <- .25

p_perception <- d_survey_agg |>
  filter(category == "perception") |>
  mutate(question_sorted = factor(question, levels = c("goalsetting","performwell","goalstress","distraction","relevance")))|>
  ggplot(aes(x = block, y = response_mean, group = interaction(exp_order, question))) +
  facet_grid(~ perception_label_sorted, labeller = labeller(question = str_to_title)) +
  geom_line(aes(lty = exp_order), position = position_dodge(width = dodge_width)) +
  geom_errorbar(aes(ymin = response_mean - response_se, ymax = response_mean + response_se, colour = condition),
                width = 0,
                alpha = .5,
                position = position_dodge(width = dodge_width)) +
  geom_point(aes(colour = condition, pch = condition),
             size = 2,
             position = position_dodge(width = dodge_width)) +
  scale_y_continuous(breaks = 1:7) +
  scale_colour_manual(values = col_condition) +
  guides(lty = "none") +
  labs(x = "Block",
       y = "Response",
       colour = "Condition",
       pch = "Condition") +
  theme_paper

p_perception
ggsave(p_perception, file = here("output", "survey_perception.png"), width = 8, height = 3)
```

## Goalsetting 

### Fit frequentist model

```{r}
m_goalsetting <- lmer(response ~ gamified +
                    gamified:exp_group_c +
                    gamified:gamified_first_c +
                    gamified:gamified_first_c:exp_group_c +
                    (1 | subject),
                  data = filter(d_survey_m, question == "goalsetting"))

summary(m_goalsetting)
print_model_table(m_goalsetting)
```

#### Fitted values

Gamified versus control:
```{r}
d_model_fit <- crossing(
  gamified = sort(unique(d_survey_m$gamified)), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalsetting,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

Points group versus progress bar group:
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalsetting,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

Control condition: gamified first vs. gamified second:
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_goalsetting,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit

```

Control condition: progress bar group vs. points group:
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalsetting,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```


#### Visualise fitted model

```{r}
p_goalsetting_m <- plot_model_fit(m_goalsetting, filter(d_survey_m, question == "goalsetting"), y_lab = "Goal use") +
  scale_y_continuous(limits = c(3, 6), labels = scales::comma_format())

p_goalsetting_m
```
### Fit Bayesian model

```{r}
m_goalsetting_bayes <- brm(response ~ gamified +
                             gamified:exp_group_c +
                             gamified:gamified_first_c +
                             gamified:gamified_first_c:exp_group_c +
                             (1 | subject),
                           data = filter(d_survey_m, question == "goalsetting"),
                           prior = set_prior("normal(0, 1)", class = "b"),
                           chains = 4,
                           iter = 11000,
                           warmup = 1000,
                           sample_prior = TRUE,
                           future = TRUE,
                           seed = 0)

summary(m_goalsetting_bayes)
```
Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_goalsetting_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```
#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_goalsetting <- hypothesis(m_goalsetting_bayes,
                                 c("gamifiedTRUE = 0",
                                   "gamifiedFALSE:exp_group_c = 0",
                                   "gamifiedTRUE:exp_group_c = 0",
                                   "gamifiedFALSE:gamified_first_c = 0",
                                   "gamifiedTRUE:gamified_first_c = 0",
                                   "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                                   "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                                 class = "b")

h_test_goalsetting$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.

```{r, fig.height = 12, fig.width = 6}
sd_ratio_goalsetting <- plot(h_test_goalsetting, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_goalsetting[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values
Gamified versus control:
```{r}
d_model_fit <- crossing(
  gamified = sort(unique(d_survey_m$gamified)), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalsetting_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

Points group versus progress bar group:
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalsetting_bayes,,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

Control condition: gamified first vs. gamified second:
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_goalsetting_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit

```

Control condition in progress bar group vs. points group:
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalsetting_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

## Wanting to perform well

### Fit frequentist model

```{r}
m_performwell <- lmer(response ~ gamified +
                        gamified:exp_group_c +
                        gamified:gamified_first_c +
                        gamified:gamified_first_c:exp_group_c +
                        (1 | subject),
                      data = filter(d_survey_m, question == "performwell"))

summary(m_performwell)
print_model_table(m_performwell)
```


#### Fitted values
Gamified versus control:
```{r}
d_model_fit <- crossing(
  gamified = sort(unique(d_survey_m$gamified)), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_performwell,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

Points group versus progress bar group:
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_performwell,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

Control condition: gamified first vs. gamified second:
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_performwell,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit

```

Control condition in progress bar group vs. points group:
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_performwell,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit

```


#### Visualise fitted model

```{r}
p_performwell_m <- plot_model_fit(m_performwell, filter(d_survey_m, question == "performwell"), y_lab = "Wanting to perform well") +
  scale_y_continuous(limits = c(4, 7), labels = scales::comma_format())

p_performwell_m
```
### Fit Bayesian model

```{r}
m_performwell_bayes <- brm(response ~ gamified +
                             gamified:exp_group_c +
                             gamified:gamified_first_c +
                             gamified:gamified_first_c:exp_group_c +
                             (1 | subject),
                           data = filter(d_survey_m, question == "performwell"),
                           prior = set_prior("normal(0, 1)", class = "b"),
                           chains = 4,
                           iter = 11000,
                           warmup = 1000,
                           sample_prior = TRUE,
                           future = TRUE,
                           seed = 0)

summary(m_performwell_bayes)
```
Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_performwell_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```
#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_performwell <- hypothesis(m_performwell_bayes,
                                 c("gamifiedTRUE = 0",
                                   "gamifiedFALSE:exp_group_c = 0",
                                   "gamifiedTRUE:exp_group_c = 0",
                                   "gamifiedFALSE:gamified_first_c = 0",
                                   "gamifiedTRUE:gamified_first_c = 0",
                                   "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                                   "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                                 class = "b")

h_test_performwell$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.


```{r, fig.height = 12, fig.width = 6}
sd_ratio_performwell <- plot(h_test_performwell, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_performwell[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values
Gamified versus control:
```{r}
d_model_fit <- crossing(
  gamified = sort(unique(d_survey_m$gamified)), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_performwell_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

Points group versus progress bar group:
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_performwell_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```

Control condition: gamified first vs. gamified second:
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_performwell_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA, 
                                 type = "response")[,1]

d_model_fit
```

Control condition in progress bar group vs. points group:
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_performwell_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```


## Conclusion
-	The two aspects of goal setting (feeling like working towards a goal, trying to perform well) were rated higher in the gamified conditions than the control condition.
- Learners felt more like they were working towards a goal with progress bar feedback than with points, though there was no difference in this regard between the two gamified conditions.
-	There were also indirect effects: Learners felt less like working towards a goal and tried less to perform well in the control condition when they had started with gamified practice and/or when in the progress bar group.


# Are there negative effects of gamified feedback on learners' experience?

Relevant variables: stress and distraction.

## Stress

### Fit frequentist model

```{r}
m_goalstress <- lmer(response ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject),
                     data = filter(d_survey_m, question == "goalstress"))

summary(m_goalstress)
print_model_table(m_goalstress)
```

#### Fitted values
Gamified versus control:
```{r}
d_model_fit <- crossing(
  gamified = sort(unique(d_survey_m$gamified)), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalstress,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```


Points group versus progress bar group:
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalstress,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

#### Visualise fitted model

```{r}
p_goalstress_m <- plot_model_fit(m_goalstress, filter(d_survey_m, question == "goalstress"), y_lab = "Goal stress") +
  scale_y_continuous(limits = c(3, 6), labels = scales::comma_format())

p_goalstress_m
```
### Fit Bayesian model

```{r}
m_goalstress_bayes <- brm(response ~ gamified +
                            gamified:exp_group_c +
                            gamified:gamified_first_c +
                            gamified:gamified_first_c:exp_group_c +
                            (1 | subject),
                          data = filter(d_survey_m, question == "goalstress"),
                          prior = set_prior("normal(0, 1)", class = "b"),
                          chains = 4,
                          iter = 11000,
                          warmup = 1000,
                          sample_prior = TRUE,
                          future = TRUE,
                          seed = 0)

summary(m_goalstress_bayes)
```
Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_goalstress_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```
#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_goalstress <- hypothesis(m_goalstress_bayes,
                                c("gamifiedTRUE = 0",
                                  "gamifiedFALSE:exp_group_c = 0",
                                  "gamifiedTRUE:exp_group_c = 0",
                                  "gamifiedFALSE:gamified_first_c = 0",
                                  "gamifiedTRUE:gamified_first_c = 0",
                                  "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                                  "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                                class = "b")

h_test_goalstress$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.


```{r, fig.height = 12, fig.width = 6}
sd_ratio_goalstress <- plot(h_test_goalstress, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_goalstress[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values
Gamified versus control:
```{r}
d_model_fit <- crossing(
  gamified = sort(unique(d_survey_m$gamified)), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalstress_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```


Points group versus progress bar group:
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_goalstress_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```


## Distraction

### Fit frequentist model
```{r}
m_distraction <- lmer(response ~ gamified +
                       gamified:exp_group_c +
                       gamified:gamified_first_c +
                       gamified:gamified_first_c:exp_group_c +
                       (1 | subject),
                     data = filter(d_survey_m, question == "distraction"))

summary(m_distraction)
print_model_table(m_distraction)
```

#### Fitted values

Points versus progress bar condition
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_distraction,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit

```

Control condition in Block 1 versus Block 2
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$exp_group_c))
)

d_model_fit$model_fit <- predict(m_distraction,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit

```


Gamified conditions in Block 1 versus Block 2
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_distraction,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit

```


#### Visualise fitted model

```{r}
p_distraction_m <- plot_model_fit(m_distraction, filter(d_survey_m, question == "distraction"), y_lab = "Distraction") +
  scale_y_continuous(limits = c(1.5, 4.5), labels = scales::comma_format())

p_distraction_m
```
### Fit Bayesian model

```{r}
m_distraction_bayes <- brm(response ~ gamified +
                             gamified:exp_group_c +
                             gamified:gamified_first_c +
                             gamified:gamified_first_c:exp_group_c +
                             (1 | subject),
                           data = filter(d_survey_m, question == "distraction"),
                           prior = set_prior("normal(0, 1)", class = "b"),
                           chains = 4,
                           iter = 11000,
                           warmup = 1000,
                           sample_prior = TRUE,
                           future = TRUE,
                           seed = 0)

summary(m_distraction_bayes)
```
Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_distraction_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```
#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_distraction <- hypothesis(m_distraction_bayes,
                                 c("gamifiedTRUE = 0",
                                   "gamifiedFALSE:exp_group_c = 0",
                                   "gamifiedTRUE:exp_group_c = 0",
                                   "gamifiedFALSE:gamified_first_c = 0",
                                   "gamifiedTRUE:gamified_first_c = 0",
                                   "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                                   "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                                 class = "b")

h_test_distraction$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.

```{r, fig.height = 12, fig.width = 6}
sd_ratio_distraction <- plot(h_test_distraction, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_distraction[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values

Points versus progress bar condition
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_distraction_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit

```

Control condition in Block 1 versus Block 2
```{r}
d_model_fit <- crossing(
  gamified = FALSE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$exp_group_c))
)

d_model_fit$model_fit <- predict(m_distraction_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```


Gamified conditions in Block 1 versus Block 2
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = 0,
  gamified_first_c = sort(unique(d_survey_m$gamified_first_c))
)

d_model_fit$model_fit <- predict(m_distraction_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit

```

## Conclusion
-	Gamified practice was rated as more stressful than the control condition.
-	The progress bar condition was rated as more stressful and more distracting than the points condition. (However, ratings were overall relatively low on the 7-point scale.
-	There was also an indirect effect in that the gamified feedback was rated as more distracting (and the timer in the control condition as less distracting) when presented in the second block.  


# Which Condition do Learners prefer?

Relevant variables: perceived relevance of gamification, preference for condition.

## Perceived Relevance

### Fit frequentist model
Since perceived relevance was only rated in the gamified conditions, we cannot use the same model as elsewhere.
Instead we can use a simpler linear regression:
```{r}
m_relevance <- lm(response ~ exp_group_c +
                    gamified_first_c +
                    gamified_first_c:exp_group_c,
                  data = filter(d_survey_m, question == "relevance", gamified == TRUE))

summary(m_relevance)
```

#### Fitted values

```{r}
d_model_fit <- crossing(
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0,
)

d_model_fit$model_fit <- predict(m_relevance,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

### Fit Bayesian model

```{r}
m_relevance_bayes <- brm(response ~ exp_group_c +
                           gamified_first_c +
                           gamified_first_c:exp_group_c,
                         data = filter(d_survey_m, question == "relevance", gamified == TRUE),
                         prior = set_prior("normal(0, 1)", class = "b"),
                         chains = 4,
                         iter = 11000,
                         warmup = 1000,
                         sample_prior = TRUE,
                         future = TRUE,
                         seed = 0)

summary(m_relevance_bayes)
```
Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 6, fig.width = 8}
plot(m_relevance_bayes, nvariables = 4, variable = "^b_", regex = TRUE)
```
#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_relevance <- hypothesis(m_relevance_bayes,
                                 c("exp_group_c = 0",
                                   "gamified_first_c = 0",
                                   "exp_group_c:gamified_first_c = 0"),
                                 class = "b")

h_test_relevance$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.


```{r, fig.height = 6, fig.width = 6}
sd_ratio_relevance <- plot(h_test_relevance, nvariables = 4, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_relevance[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

#### Fitted values

```{r}
d_model_fit <- crossing(
  exp_group_c = sort(unique(d_survey_m$exp_group_c)),
  gamified_first_c = 0,
)

d_model_fit$model_fit <- predict(m_relevance_bayes,
                                 newdata = d_model_fit,
                                 re_formula = NA,
                                 type = "response")[,1]

d_model_fit
```



## Preference for Condition

Contingency table: what proportion of participants in each condition preferred the gamified practice over the control condition or indicated no preference?
```{r}
data_preferences <- d_survey |> 
  filter(preference!="" & condition != "Control") |> # Exclude two participants who did not respond to this item and keep only one row per participant
  group_by(exp_group = as.factor(exp_group), subject) |>
  summarise(preference = as.factor(first(preference)), .groups = "drop")

# Counts
crosstab_preferences <- xtabs(~exp_group + preference, data = data_preferences)
crosstab_preferences

# Proportions
prop.table(crosstab_preferences,1)
```

### Frequentist Chi-square test
```{r}
chisq.test(crosstab_preferences)
```
### Bayesian Chi-square test
```{r}
bf_preferences <- contingencyTableBF(crosstab_preferences, sampleType = "indepMulti", fixedMargin = "rows")
1/bf_preferences
```

## Conclusions
-	Progress bar condition was rated as more relevant than the points condition. 
- However, participants in the progress bar group did not show a stronger preference for gamified practice over the control condition than participants in the points group.


# Does gamification change learners' metacognitive judgements?

Relevant variable: judgement of learning.

## Judgement of learning
Participants were asked to give the percentage of practiced translations they thought they would still know in two days.

```{r}
d_jol_agg <- d_survey |>
  group_by(subject, block, condition, gamified, gamified_first, exp_group) |>
  summarise(judgement_of_learning = judgement_of_learning[1]/100) |>
  group_by(block, condition, gamified, gamified_first, exp_group) |>
  summarise(jol_mean = mean(judgement_of_learning, na.rm = T),
            jol_se = sd(judgement_of_learning, na.rm = T)/sqrt(n())) |>
  ungroup() |>
  add_experiment_cols()


dodge_width <- .25

p_jol <- ggplot(d_jol_agg, aes(x = block, y = jol_mean, group = exp_order)) +
  geom_line(aes(lty = exp_order), position = position_dodge(width = dodge_width)) +
  geom_errorbar(aes(ymin = jol_mean - jol_se, ymax = jol_mean + jol_se, colour = condition),
                width = 0,
                alpha = .5,
                position = position_dodge(width = dodge_width)) +
  geom_point(aes(colour = condition, pch = condition),
             size = 2,
             position = position_dodge(width = dodge_width)) +
  scale_colour_manual(values = col_condition) +
  scale_y_continuous(labels = scales::percent_format()) +
  guides(lty = "none") +
  labs(x = "Block",
       y = "Judgement of learning",
       colour = "Condition",
       pch = "Condition") +
  theme_paper

p_jol
ggsave(p_jol, filename = here("output", "survey_judgementoflearning.png"), width = 8, height = 3)
```

### Fit frequentist model

```{r}
d_jol_m <- d_survey_m |>
  group_by(subject, gamified, block, condition, gamified_first, exp_group, gamified_first_c, exp_group_c) |>
  summarise(judgement_of_learning = judgement_of_learning[1]/100) |>
  ungroup()


m_jol <- lmer(judgement_of_learning ~ gamified +
                gamified:exp_group_c +
                gamified:gamified_first_c +
                gamified:gamified_first_c:exp_group_c +
                (1 | subject),
              data = d_jol_m)

summary(m_jol)
print_model_table(m_jol)
```

#### Fitted values
Gamified versus control:
```{r}
d_model_fit <- crossing(
  gamified = sort(unique(d_jol_m$gamified)), 
  exp_group_c = 0,
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_jol,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```


Points group versus progress bar group:
```{r}
d_model_fit <- crossing(
  gamified = TRUE, 
  exp_group_c = sort(unique(d_jol_m$exp_group_c)),
  gamified_first_c = 0
)

d_model_fit$model_fit <- predict(m_jol,
                                 newdata = d_model_fit,
                                 re.form = NA, 
                                 type = "response")

d_model_fit
```

### Visualise fitted model
```{r}
p_jol_m <- plot_model_fit(m_jol, d_jol_m, y_lab = "Judgement of learning") |
    scale_y_continuous(limits = c(.25, .45), labels = scales::percent_format())

p_jol_m
```
### Fit Bayesian model

```{r}
m_jol_bayes <- brm(judgement_of_learning ~ gamified +
                     gamified:exp_group_c +
                     gamified:gamified_first_c +
                     gamified:gamified_first_c:exp_group_c +
                     (1 | subject),
                   data = d_jol_m,
                   prior = set_prior("normal(0, 1)", class = "b"),
                   chains = 4,
                   iter = 11000,
                   warmup = 1000,
                   sample_prior = TRUE,
                   future = TRUE,
                   seed = 0)

summary(m_jol_bayes)
```
Inspect the posterior sample distributions of the fixed effects:
```{r, fig.height = 12, fig.width = 8}
plot(m_jol_bayes, nvariables = 8, variable = "^b_", regex = TRUE)
```
#### Bayes factors

Do a hypothesis test for all fixed-effect coefficients (both main effects and interactions) in the model being equal to zero.
The column `Evid.Ratio` shows the Bayes Factor in favour of the null hypothesis ($BF_{01}$).
```{r}
h_test_jol <- hypothesis(m_jol_bayes,
                         c("gamifiedTRUE = 0",
                           "gamifiedFALSE:exp_group_c = 0",
                           "gamifiedTRUE:exp_group_c = 0",
                           "gamifiedFALSE:gamified_first_c = 0",
                           "gamifiedTRUE:gamified_first_c = 0",
                           "gamifiedFALSE:exp_group_c:gamified_first_c = 0",
                           "gamifiedTRUE:exp_group_c:gamified_first_c = 0"),
                         class = "b")
h_test_jol$hypothesis |>
  mutate(BF10 = 1 / Evid.Ratio,
         evidence_for_null = sapply(Evid.Ratio, bf_to_strength))
```

This hypothesis test is calculating the Savage-Dickey density ratio at zero, which is a ratio of the posterior density at zero relative to the prior density at zero (indicated by dashed vertical line).
Values above 1 indicate a stronger belief that the effect is indeed zero after having observed the data.

```{r, fig.height = 12, fig.width = 6}
sd_ratio_jol <- plot(h_test_jol, nvariables = 8, variable = "^b_", regex = TRUE, plot = FALSE)

sd_ratio_jol[[1]] +
  geom_vline(xintercept = 0, linetype = "dashed", colour = "grey25")
```

## Conclusion

-	The frequentist analysis finds that judgements of learning were higher in the gamified conditions than in the control condition, but not significantly different between the points and progress bar condition. The Bayesian analysis finds evidence against all effects.


# Save hypothesis testing output for visualisation
```{r}
fwrite(h_test_competence$hypothesis, here("output", "hypothesis_tests", "h_test_competence.csv"))
fwrite(h_test_enjoyment$hypothesis, here("output", "hypothesis_tests", "h_test_enjoyment.csv"))
fwrite(h_test_value$hypothesis, here("output", "hypothesis_tests", "h_test_value.csv"))
fwrite(h_test_goalsetting$hypothesis, here("output", "hypothesis_tests", "h_test_goalsetting.csv"))
fwrite(h_test_performwell$hypothesis, here("output", "hypothesis_tests", "h_test_performwell.csv"))
fwrite(h_test_goalstress$hypothesis, here("output", "hypothesis_tests", "h_test_goalstress.csv"))
fwrite(h_test_distraction$hypothesis, here("output", "hypothesis_tests", "h_test_distraction.csv"))
fwrite(h_test_relevance$hypothesis, here("output", "hypothesis_tests", "h_test_relevance.csv"))
fwrite(h_test_jol$hypothesis, here("output", "hypothesis_tests", "h_test_jol.csv"))
```


# Session info
```{r}
sessionInfo()
```